A Riemannian Optimization Approach to Clustering Problems

This paper considers the optimization problem in the form of minX ∈ Fv f(x) + λ |X|1, where f is smooth, Fv = X ∈ ℝn × q : XT X = Iq, v ∈ span(X), and v is a given positive vector. The clustering models including but not limited to the models used by k-means, community detection, and normalized cut can be reformulated as such optimization problems. It is proven that the domain Fv forms a compact embedded submanifold of ℝn × q and optimization-related tools including a family of computationally efficient retractions and an orthonormal basis of any normal space of Fv are derived. An inexact accelerated Riemannian proximal gradient method that allows adaptive step size is proposed and its global convergence is established. Numerical experiments on community detection in networks and normalized cut for image segmentation are used to demonstrate the performance of the proposed method.